# Singular Matrix

A singular matrix is a square matrix of determinant “0.” i.e., a square matrix A is singular if and only if det A = 0. Inverse of a matrix A is found using the formula A-1 = (adj A) / (det A). Thus, a matrix is called a square matrix if its determinant is zero. Now let us discuss about singular matrix, its properties, and others in detail.

## What is a Singular Matrix?

A square matrix is said to be a singular matrix if its determinant is zero and it is not invertible. In a singular matrix, some rows and columns are linearly dependent. As the rows and columns of a singular matrix are linearly dependent, the rank of the matrix will be less than the order of the matrix.

The image given below is an “m Ã— n” matrix that has “m” rows and “n” columns.

We know that the formula to determine the inverse of a matrix is equal to the adjoint of the matrix divided by the determinant of the matrix, i.e., A-1 = (adj A) / |A|. From the definition of a singular matrix, we know that |A| = 0, so its inverse is not defined.

Let us consider that A and B are two square matrices of order “n Ã— n”

If,

AB = BA = I

where,

• I is an identity or unit matrix of order n
• B is said to be the inverse matrix of A

Thus, matrix A is a non-singular matrix.

## Properties of a Singular Matrix

The following are the properties of the Singular Matrix:

• Every singular matrix must be a square matrix, i.e., a matrix that has an equal number of rows and columns.
• The determinant of a singular matrix is equal to zero.
• As the determinant of a singular matrix is zero, its inverse is not defined.
• A zero matrix of any order matrix is a singular matrix, as its determinant is zero.
• In a singular matrix, some rows and columns are linearly dependent.
• The rank of a singular matrix will be less than the order of the matrix, i.e., Rank (A) < Order of A.
• A matrix that has any two rows or any two columns identical is singular, as the determinant of such a matrix is zero.
• When a row or column’s elements in a matrix are all zeros, then the matrix is singular, as its determinant is zero.
• When one row (or column) of a matrix is a scalar multiple of another row (or column), then the matrix is singular as its determinant is zero.Â

## Differences Between Singular and Non-Singular Matrix

Differences between Singular Matrix and Non-Singular Matrix can be understood using the table given below

Singular Matrix Vs Non-Singular Matrix

Â Singular MatrixÂ

Â Non-Singular MatrixÂ

A square matrix is said to be a singular matrix if its determinant is zero, i.e., det A = 0.

A square matrix is said to be a non-singular matrix if its determinant is not zero, i.e., det A â‰  0.

If a matrix is singular, then its inverse is not defined.

If a matrix is non-singular, then its inverse is defined.

The rank of a singular matrix will be less than the order of the matrix, i.e., Rank (A) < Order of A.

The rank of a non-singular matrix will be equal to the order of the matrix, i.e., Rank (A) = Order of A.

In a singular matrix, some rows and columns are linearly dependent.

In a non-singular matrix, all the rows and columns are linearly independent.

[Tex]A = \left(\begin{array}{ccc} 2 & 2 & 4\\ 1 & 1 & 2\\ 3 & 7 & 9 \end{array}\right) [/Tex]

[Tex]B = \left[\begin{array}{ccc} 1 & 2 & -3\\ 6 & 0 & 8\\ -1 & 4 & 0 \end{array}\right] [/Tex]

## Identifying a Singular Matrix

Follow the conditions given below to determine whether the given matrix is singular or not.

• Determine whether the given matrix is a square matrix or not.
• If the given matrix is a square matrix, then find the determinant of the matrix.

â‡’ If |A|= 0, then the given matrix is singular.

â‡’ If |A|â‰ 0, then the given matrix is non-singular.

### Formula for Determinant of “2 Ã— 2” Matrix

If A =Â [Tex]\left[\begin{array}{cc} a & b\\ c & d \end{array}\right]Â Â Â Â  [/Tex]Â is a “2 Ã— 2” matrix, then its determinant isÂ

### Formula for Determinant of “3 Ã— 3” Matrix

If A =Â [Tex]\left[\begin{array}{ccc} a_{1} & a_{2} & a_{3}\\ b_{1} & b_{2} & b_{3}\\ c_{1} & c_{2} & c_{3} \end{array}\right]Â Â Â Â  [/Tex]Â is a “3 Ã— 3” matrix, then its determinant isÂ

|A|= a1(b2c3 â€“ b3c2) â€“ a2(b1c3 â€“ b3c1) + a3(b1c2 â€“ b2c1)

Also, Check

## Solved Examples on Singular Matrix

Example 1: Find the value of k if the matrix given below, is a singular matrix.

[Tex]A = \left[\begin{array}{cc} k & -4\\ 5 & 2 \end{array}\right] [/Tex]

Solution:

Given matrix A =Â [Tex]\left[\begin{array}{cc} k & -4\\ 5 & 2 \end{array}\right] [/Tex]

We know that the determinant of a singular matrix is zero, i.e., det A = 0

â‡’ (2Ã—k) â€“ (â€“4 Ã— 5) = 0

â‡’ 2k + 20 = 0

â‡’ 2k = -20

â‡’ k = â€“20/2 = â€“10

Hence, the value of k if the given matrix is a singular matrix is â€“10.

Example 2: Determine the inverse of the matrix given below.

[Tex]P = \left[\begin{array}{cc} -3 & 4\\ 6 & -8 \end{array}\right] [/Tex]

Solution:

Given matrixÂ [Tex]Â P = \left[\begin{array}{cc} -3 & 4\\ 6 & -8 \end{array}\right] [/Tex]

P-1 = Adj P / |P|

Now, let us find the determinant of the matrix P.

|P| = (â€“3 Ã— â€“8) â€“ (6 Ã— 4)

|P| = 24 â€“ 24 = 0

Since, the determinant of matrix P = 0, it is a singular matrix, and its inverse matrix doesn’t exist.

Example 3: Determine whether the given matrix is singular or not.

[Tex]A = \left[\begin{array}{ccc} 1 & 0 & -3\\ 0 & 5 & 2\\ -1 & 4 & 0 \end{array}\right] [/Tex]

Solution:

Given matrix A =Â [Tex]\left[\begin{array}{ccc} 1 & 0 & -3\\ 0 & 5 & 2\\ -1 & 4 & 0 \end{array}\right] [/Tex]

To determine whether the given matrix is singular or not, we have to find its determinant.

det A = 1[(5 Ã— 0) â€“ (4 Ã— 2)] â€“ 0[(0 Ã— 0) â€“ (2 Ã— â€“1)] + (-3) [(0 Ã— 4) â€“ (â€“1 Ã— 5)]

â‡’ |A| = (1 Ã— -8) â€“ 0 + (â€“3 Ã— 5)Â

â‡’ |A| = â€“8 â€“ 15 = â€“23 â‰  0

Since the determinant of the given matrix is not equal to zero, it is a non-singular matrix.

Example 4: Find the value of b if the matrix given below, is a singular matrix.

[Tex]B = \left[\begin{array}{cc} 9 & b\\ 6 & -2 \end{array}\right] [/Tex]

Solution:

Given matrixÂ [Tex]B = \left[\begin{array}{cc} 9 & b\\ 6 & -2 \end{array}\right] [/Tex]

We know that the determinant of a singular matrix is zero, i.e., det B = 0

â‡’ (9 Ã— â€“2) â€“ (6 Ã— b) = 0

â‡’ â€“18 â€“ 6b = 0

â‡’ â€“6b = 18

â‡’ b = 18/â€“6 = â€“3

Hence, the value of b if the given matrix is a singular matrix is â€“3.

## FAQs on Singular Matrix

### 1. Define a Matrix.

A matrix is defined as a rectangular array of numbers that are arranged in rows and columns.

### 2. What is a Singular Matrix?

A square matrix is said to be a singular matrix if its determinant is zero and it is not invertible.

### 3. What is the Rank of a Singular Matrix of Order “3 Ã— 3”?

If the given matrix A is singular, then its determinant is zero. Now, the rank of the given matrix will be less than the order of the matrix, i.e., rank (A) < 3.

### 4. What is the Determinant of a Singular Matrix?

The determinant of a matrix determines whether it is singular or non-singular. So, a matrix is said to be singular if its determinant is zero.

### 5. Is a Zero Matrix a Singular Matrix?

As the determinant of a singular matrix is zero, it is a singular matrix.

### 6. What is Rank of a Singular Matrix?

The rank of singular matrix ‘n’ is always less than ‘n’.

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